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It is not difficult to prove that the union of a chain (or, more generally, a directed family) of subspaces of a vector space $V$ is a subspace of $V$.

Given a family $\mathcal{F}$ of subspaces of a vector space $V$ such that the union of $\mathcal{F}$ is a subspace of $V$, is it true that $\mathcal{F}$ is a directed family?

If not, is there a "nice" characterization of families of subspaces whose union is a subspace?

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    The union of two subspaces is a subspace if and only if one of the subspaces is contained in the other. – 2017-01-04
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    Is it a finite family? – 2017-01-04
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    For a vector space over an infinite field, it is never the case that a finite family unions to a subspace, unless one of the family's elements contains all the others. – 2017-01-04

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The family $\mathcal{F}$ of one-dimensional subspaces of $V$ has the property that its union is the whole space, but it is not directed. Note that if $V$ is finite-dimensional over a finite field, then $\mathcal{F}$ is finite as well. I don't know of any characterisation though.